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The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities. [ABSTRACT FROM AUTHOR]

We consider rational surfaces Z defined by divisorial valuations ν of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when ν is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given. [ABSTRACT FROM AUTHOR]